Problem 20: Rigid Motions and Kinematic Density

I. The Invariance of Kinematic Density

The wedge product of , , and is , where is the expression for kinematic density, which will be defined in more detail a bit later.

Let's prove the invariance of . Because , , and are all left-invariant, it follows that is a left-invariant 3-form. In fact, it is the unique left-invariant 3-form. Suppose we have

Because is left-invariant,

and implies that must be a constant function. (A similar case was discussed in Problem 19). So

is left-invariant. A similar argument shows that

is also invariant under right translations. So, is both left-invariant and right-invariant. Using the fact that , it follows that

II. Kinematic Measure

In the past we have discussed geometric probability in terms of "throwing points at" or "dropping lines on" a region in order to determine information regarding its length or area. Now let's consider probability in a different light. Suppose we have region K (K need not be convex) and a line L. Let's measure the set of motions that will move that line to a unique line that meets the region K. Wouldn't the set of lines that intersect K (Figure 1) have the same measure as the set of motions that move a line L so that it intersects K (Figure 2) ? YES!!

The kinematic density is an element of volume, and integrating over a domain on M yields the measure of the set of motions called kinetic measure.

The benefit of measuring a set of motions rather than a set of lines is that we are no longer restricted to just lines. We can calculate probability by measuring the motions of any figure we choose, like rectangles for example. Let K = OABC be a rectangle and let be a fixed region on the plane. Let's find the measure of the set of motions that move K so that it intersects in at least one point. This measure is defined by the integral of over all the motions that satsify . Note that the measure of the set of motions does not depend on the initial position of our rectangle or the position of the region .

Because is left-invariant, we can also measure the set of motions such that , where . Then . It is also true that . The figure below illustrates examples of possible images of and ; it is easy to see that the measure is the same in both cases and that all three sets are congruent.

PROBLEM 20:

If the rectangle K reduces to a single point and we define , what is the measure of the set of motions that send the point to the region . In other words, find .